Table Of ContentRapid Cooling to Quantum Degeneracy in Dynamically Shaped Atom Traps
Richard Roy, Alaina Green, Ryan Bowler, and Subhadeep Gupta
Department of Physics, University of Washington, Seattle, Washington 98195, USA
(Dated: April22, 2016)
We report on a general method for the rapid production of quantum degenerate gases. Using
174Yb, we achieve an experimental cycle time as low as (1.6−1.8)s for the production of Bose-
Einstein condensates (BECs) of (0.5−1)×105 atoms. While laser cooling to 30µK proceeds in a
standard way, evaporative cooling is highly optimized by performing it in an optical trap that is
6
dynamically shaped by utilizing the time-averaged potential of a single laser beam moving rapidly
1
in one dimension. We also produce large (>106) atom number BECs and successfully model the
0
2 evaporationdynamicsovermorethanthreeordersofmagnitudeinphasespacedensity. Ourmethod
providesasimpleandgeneralapproachtosolvingtheproblemoflongproductiontimesofquantum
r degenerate gases.
p
A
1 I. INTRODUCTION magnetic atoms and liberating the magnetic degree of
2 freedom for interaction control during cooling [12, 13].
Standard ODTs have small volumes and high collision
Theproductionofquantumdegenerategaseshasrevo-
h] lutionizedthefieldofatomicphysics. Suchgasesarenow rates,leadingtosmalleratomnumbersbutshorterevap-
orationtimescalesintherangeofafewto10s. Anoverall
p routinely used as a means towards understanding com-
- plexmany-bodyquantumphenomenafromtherealmsof BECproductiontimeaslowas2shasbeenaccomplished
m condensed matter and nuclear physics [1, 2]. As atom in an ODT [14] by combining one broad and one very
o sources with precisely controlled properties, these gases narrow transition for laser cooling in 84Sr, to realize an
at can also significantly advance applications such as atom extremelyhighinitialPSDof0.1beforeevaporativecool-
. interferometry [3] and quantum information processing ing[15]. ABECproductiontime of3.3swasachievedin
cs [4, 5]. While the production and measurement meth- 87Rb by combining sub-Doppler laser cooling to a high
i odsofquantumgasexperimentsarewellestablished,the initialPSDof2×10−3withfastevaporativecoolinginan
s
y measurement rate remains substantially limited by the ODT in which the volume was dynamically compressed
h lack of a general method for rapid sample production. by a moving lens [16]. These methods however are not
p Cycle times for such experiments are dominated by the easilyadaptableinageneralwayto otheratomicspecies
[ production time, typically tens of seconds, while the ac- and experimental setups.
4 tualexperimentonthepreparedsamplelastsforabouta In this paper we present a general technique for rapid
v second before destructive measurement. This separation quantum degenerate gas production, where evapora-
3 of timescales is a severe impediment to the employment tive cooling is optimized by dynamically controlling the
0 of quantum degenerate gases towards precision devices ODT shape with the time-averaged potential of a sin-
1
such as atomic clocks, inertial sensors and gravimeters gle rapidly-movinglaserbeam. This method straightfor-
5
[3, 6, 7]. Bridging these timescales can significantly con- wardly allows for large initial and small final trap vol-
0
. tribute to all classes of quantum gas explorations and umes, combining the advantages of earlier evaporative
1 applications,asmostmeasurementsrelyonthestatistics cooling strategies. It is applicable to all atoms and re-
0
of results from many experimental iterations. quires no hardware beyond a standard optical trapping
6
The root of this timescale problem lies in the speed of setup. Applied to bosonic 174Yb with modest laser cool-
1
: collisionalevaporative cooling. In a standard degenerate ingtoPSDs<10−4,ourexperimentproducesBECscon-
v gas production sequence, the initial step of laser cooling taining(0.5−1)×105atomswithanoverallcycletimeof
i
X producestemperaturesinthefewtensofµK,whilelight- (1.6−1.8)s. Bysuitablyalteringthetime dependenceof
r inducedprocesseskeepthedensitybelow1012cm−3. The the trap parameters, we also produce large 174Yb BECs
a resultingphasespacedensity(PSD)of10−5−10−4 isin- with1.2×106atoms. Theobservedevaporationdynamics
creased to quantum degeneracy by subsequent evapora- are successfully captured over three orders of magnitude
tive cooling in either magnetic or optical traps. Typical in PSD by our theoretical model.
magnetictrapshavelargevolumesandrelativelylowini- Forced evaporative cooling works by removing the
tialdensities,yieldinglowcollisionratesandlongevapo- high-energy tail of the Maxwell-Boltzmann distribu-
rativecoolingtimescales oftens of seconds. The produc- tion and allowing the remaining atomic sample to re-
tion of BECs in small-volume magnetic chip traps has equilibrate by elastic collisions to a lower temperature.
provided one solution to the timescale problem [8, 9]. Keeping the removal point fixed at ηk T relative to the
B
While this method has recently achieved cycle times of temperature T allows the derivation of scaling laws for
onesecondwith87Rb[10],itcanonlybeappliedtomag- the evaporation dynamics [17, 18]. The rate of elastic
netic atoms. collisions Γ = n σv¯ and η determine the per-particle
el 0
Optical dipole traps (ODTs)[11, 12] provide the flexi- evaporative loss rate as Γ = Γ (η − 4)e−η for large
ev el
bility to cool all atoms, enabling applications with non- η [18, 19], where n is the peak particle density, σ is
0
2
(a) scaling of R with density means maintaining a large N
leads to lowered ω¯, reduced collision rates and longer
Probe 1ω
R evaporation timescales. Numerical modeling of evapora-
M2 65 L1 M1 0F(t)-ω tive cooling in an ODT [22, 23] can help optimize γ, but
L2 z y 0 ̟/ω -1,0RF the challenges of large number and speed remain.
mod Oursolutiontothesechallengesinvolvestheimplemen-
A
O
λ/2 x M tation of the time-averaged optical potential of a laser
L3 Vacuum From Laser beam moving rapidly, or “painting”, in one dimension.
Chamber
(b) M3 (c) The ability to dynamically control the center position
Units of )010..05 52010wwww0000 al reduction0.1124 amiotvsfoeudtrnoubctlataoilttoihpnoontowhf(eeCtritP,mrrMaeep,s)uadaletmkspeptyihnliati(udnUdvdaee=npotefaηntgkdheBeeoTnbvt)e,earaamnrmbdeiinttfrrhaaeoqdrdyudseictnwoiocniytnthrtaooasl
u n
Ux() ( 0.0 Fractio 24 Uω/xU/ω0x,0 fixTedheporwesetr-olfawthreelpaatpioenrsihsiporbgeatnwizeeednaUsafonldlowω¯s[.18S,e2ct3i]o.n
-10 0 10 0 2 4 6 8 10 II consists of the experimental setup and our model for
x (units of w0) CPM amplitude (units of w0) the CPM trap shape. In Section III we present our nu-
merical model for forced evaporative cooling in this trap
FIG. 1: (a) Schematic of the optical trap setup. Gravity along with a procedure for optimizing evaporation effi-
points along y. The horizontally oriented AOM is driven by ciency, and apply our model to an experimentally opti-
an FM waveform (inset) resulting in parabolic intensity pro- mized evaporation trajectory. In Section IV we describe
file. Arrows on the laser beam indicate direction of CPM. rapid BEC production using our method, while in Sec-
(b) Evolution of time-averaged trap shape U(x) = U(x,0,0)
tionVwefocusontheproductionoflargeBECs. Finally,
from Gaussian to parabolic with increasing CPM amplitude
Section VI provides a summary of this technique and an
for fixed laser power in units of unmodulated beam waist w0 outlook for applications.
and trap depth U0. (c) Fractional reduction of trap depth
andfrequencyinthepaintingdirectionωx fromunmodulated
values U0 and ωx,0 for fixedpower.
II. EXPERIMENTAL SETUP
the scattering cross section, v¯ = 8k T/πm, and m is Weperformourexperimentintheapparatusdescribed
B
the particle mass. We define the evaporation efficiency in [24] using 174Yb bosons. Our laser cooling procedure
as γ = − ln(ρf/ρi) , where ρ pand N are the fi- can produce 108 atoms in 5s at 30µK in a compressed
ln(Nf/Ni) f(i) f(i) magneto-opticaltrap(MOT) operatingon the 1S →3P
nal(initial) PSD and particle number, respectively. In 0 1
transition. In the rest of this section, we describe the
the absence of additional loss processes, γ can be made
setup of our ODT and characterization of the parabolic
arbitrarilyhighbyusingalargeηandthusalongcooling
CPM trap.
timescale. Therealityofotherlossprocessestempersthis
ideaandintroducesanewtimescalewhichcompeteswith
thatofevaporativecooling. Thiscompetitionoutlinesan
importantexperimentalchallengeandiscapturedbythe A. Optical trap details
ratio of “good” to “bad” collisions R = Γ /Γ . The
el loss
prescriptionforoptimumefficiency howevercruciallyde- The ODT (Fig.1) is generated by sending the output
pends on the nature of the dominant loss process. of a fiber laser at 1064 nm (IPG YLR-100-LP) through
For 1-body loss dominated systems characteristic of an acousto-optic modulator (AOM, 80 MHz, Intraaction
standard magnetic traps, R is proportional to the elas- ATM-804DA6B) and focusing the diffracted beam (1st
tic collision rate and scales as Nω¯3T−1, where we have order) to a Gaussian waist of 35µm at the atoms. This
assumed a 3D harmonic trap with (geometric) mean lightis then refocusedwith orthogonalpolarizationback
◦
trap frequency ω¯. Comparing against the scaling ρ ∝ onto the atoms with a waistof 30µm,at an angle of 65
Nω¯3T−3, we find that maintaining or increasingthe col- with respect to the first pass.
lision rate at every step and achieving “runaway” evap- To implement the trapcenter positionmodulation, we
oration is an excellent prescription for efficient cooling modulate the center frequency of the voltage controlled
[17]. Importantly, this prescription simultaneously im- oscillator(VCO, Minicircuits ZOS-100)supplying RF to
provesthe speedofevaporativecoolingandfinalparticle theAOMat10kHzwiththewaveformshowninFig.1(a),
number. In an ODT, while 3-body processes can be ne- which results in a nearly perfect parabolic trap shape
glected in certain situations [18, 20, 21], it is often the (see Appendix). The largestCPMamplitude used is 260
dominant loss mechanism. Then R ∝ N−1ω¯−3T2, and µm (520 µm peak-to-peak), and corresponds to shifting
γ and R cannot be simultaneously optimized. Crucially, the center frequency of the AOM by 7 MHz (14 MHz
unlike in 1-body loss dominated systems, the inverted peak-to-peak) on top of 80 MHz. The orientation of
3
5 6
25x10
4 h= 89 µm r K) 24
h= 113 µm e µ10
3 h= 130 µm mb 20 U1/kB = 333 µK p ( 4
) m
Hz 2 hh== 128519 µµmm nu 15 Te 12
(
̟ m 4
2 0 5 10 15 20
ω/ to 10 U2/kB = 559 µK
a
100
b
89 Y 5 U3/kB = 12.7 µK
7
6 0
5 0 5 10 15 20
8 2 3 4 5 6 78 2 3 4 5 6
1 10 Hold Time (s)
Power per beam (W)
FIG. 3: Measurements of Yb number and temperature evo-
FIG. 2: Comparison of CPM trap model (open triangles) lution for trap depth model calibration. The 3 different trap
to measurements (solid circles) of the trap frequency. Dif- configurations are as follows: (1, solid circles) P = 58 W,
ferent CPM amplitudes are indicated with different colors. h = 259 µm; (2, solid squares) P = 58 W, h = 130 µm;
Boxes indicate trap configurations used for measurements (3, solid triangles) P = 2.2 W, h = 113 µm. The lines are
in Fig. 3. The black dashed line is a parametric plot of fitsusingtheevaporationmodeldiscussedinSectionIII.The
(ω¯(P(t),h(t))/2π,P(t))correspondingtotheevaporationtra- extracted trap depth for each curve is indicated on the plot
jectory used in Fig. 5. above. Our time-averaged trap model predicts U1/kB =300
µK, U2/kB =576 µK, and U3/kB =13.3 µK.
the AOM is such that the CPM occurs in the horizon-
tal plane. We control the overall ODT power with the To arrive at a reliable model for our specific time-
RF drive strength to the AOM. Using a power of 70W averagedpotential,wemeasurethe physicalamplitudeh
andCPM amplitude of260µmat the atoms,we capture (inµm)ofthecenterpositionmodulationattheODTfo-
up to 5×107 atoms from the compressed MOT into the cusforagivenappliedvoltagetotheVCO.Furthermore,
ODT. we account for the effects of ellipticity and thermal lens-
ingbyallowingpowerdependentwaistsw =w (P)and
x x
w = w (P), changing the arguments of the fractional
y y
B. CPM trap characterization reduction functions to fU(h/wx(P)) and fω(h/wx(P)),
and appropriately altering the equations in (1).
Crucial to the implementation of our forced evapora- Finally, we include the second pass of the crossed
tive cooling model (see Section III) is an accurate model dipole trap by using a magnification of 5/6 given by
for the optical trap shape as a function of laser power the lenses used to reimage the beam back onto the
◦
P and CPM amplitude h [25]. Without CPM and ne- atoms at 65 . We write Utotal(x,y,z) = U1(x,y,z) +
′ ′ ′
glecting gravity, one can apply the scalings U(P) ∝ P U2(x,y ,z )+mgy, where the subscripts 1 and 2 refer
and ω¯(P) ∝ P1/2 in the absence of beam imperfections to the first and second passes of the beam, and the co-
′ ′ ′ ◦
(e.g. thermallensing,astigmatism,etc.). However,asde- ordinates (x,y ,z ) are relatedto (x,y,z) by a 65 rota-
picted in Fig. 1, CPM allows for a large range of depths tion in the x−z plane. We then compute the roots of
and frequencies at each power. We calculate the trap ∂Utotal(0,y,0)/∂y|y=y∗ = 0 for a densely spaced grid of
frequencies and trap depth for a single ODT beam with P and h values, and use these roots to compute U(P,h)
∗
CPM in the x direction and traveling in the z direction and ω¯(P,h) referenced to the equilibrium position y .
as The comparison of our trap model to trap frequency
measurements made using 174Yb is shown in Fig. 2. As
8αP 4αP
ω2 = f (h/w ), ω2 = f (h/w ), discussed above, all trap parameters are independently
x πmw4 ω 0 z πmw2z2 U 0 determined by observations of the beam itself, so there
0 0 R
8αP 2αP arenofitparametersusedhere. Wefindgoodagreement
ω2 = f (h/w ), U = f (h/w ), (1)
y πmw4 U 0 πw2 U 0 between the theoretical and experimental values.
0 0
To compare our model for the trapdepth with the ex-
where α is the atomic polarizability at the ODT wave- periment, we perform measurements of Yb number and
length λ and z = πw2/λ is the Rayleigh range. temperature evolution for three different fixed settings
R 0
f (h/w ) andf (h/w ) arethe fractionalreductionfac- of ODT power and CPM amplitude (indicated in Fig. 2
ω 0 U 0
tors shown in Fig. 1(c), conveniently written as a func- withboxes). Wethenfittheobserveddynamicswithour
tion of the CPM amplitude in units of beam waist. evaporation model (see Section III) for fixed power and
4
CPMamplitude,usingthetrapdepthasafittingparam- h(t) = max(h − βt,0), and concave functions h(t) =
0
eter. TheresultsareshowninFig. 3. Thepredictedtrap max(h0−h1(1−et/τh),0)andh(t)=max(h0−(t/τh)2,0).
depth values from our model are given in the caption to For the above functions, the parameters P and h are
0 0
Fig. 3, and are in good agreement with those extracted fixed as they correspond to the trap loading conditions,
from number and temperature dynamics. while the remaining parameters are variedas part of the
optimization algorithm.
III. EVAPORATION MODEL Our optimization algorithm utilizes the gradient as-
cent method where γ = − ln(ρ(tf)/ρ(0)) is the quantity
ln(N(tf)/N(0))
We model the number and temperature evolution in to be maximized, and tf satisfies ρ(tf) = 1. For the
the high η limit, where the equation of state is well ap- power ramp profile, we find that the optimization pro-
proximated by E = 3Nk T. The dynamical equations cedure pushes the bi-exponentialprofile towardsa single
B
are then [18, 19, 22] exponential (i.e. τP1 → τP2). CPM ramp optimization
suggests that the system is very robust to the form of
N˙ =−(Γev+Γ3b+Γbg)N (2) h(t). In fact, if we use P(t) = P0e−t/τP with τP = 1s,
Γ Γ ω¯˙ Γ E and run the optimization algorithm for each of the pro-
T˙ =− ev (η+α−3)− 3b − T + sc r, (3) posedfunctionsh(t)above,theoptimizedvaluesγ are
3 3 ω¯ 3k opt
(cid:18) (cid:19) B all within 1% of each other. For these reasonswe choose
where Γ = K N−1 n3d3~r is the per-particle loss to adopt the single exponential power ramp and linear
3b 3
rate for 3-body inelastic loss, K is a temperature- CPM ramp, as these are the simplest options.
3
R
independent 3-body inelastic loss rate coefficient, α =
We find the system to be extremely robust to the
(η−5)/(η−4),andΓ andE arethe spontaneousscat-
sc r timescale of evaporation, as indicated in Fig. 4. For
tering rate and recoil energy for 174Yb in our 1064 nm
these simulations, we fix the timescale τ and optimize
ODT. For the ODT intensities used here, heating from P
the slope of the CPM ramp β. We restrict τ & 0.8, as
spontaneousscatteringissmall. Thebackgroundlifetime P
Γ−1 is independently measured to be 35s. we believe that ourmodel cannotaccurately capture the
bg evaporation dynamics on timescales faster than this due
to complications involving thermal lensing and decou-
pling ofhorizontalandverticaltemperatures. As seenin
A. Optimization of evaporation trajectory
Fig. 4, the optimized evaporation efficiency γ varies
opt
little over the range 0.8 ≤ τ ≤ 3. In fact, the slight
P
Byneglectingtheeffectsof3-bodyinelasticloss,back- slope of γ versus τ is caused by the introduction of
opt P
ground gas collisions, and spontaneous scattering, one a new timescale, the background lifetime Γ−1. As de-
can construct analytical solutions to equations (2)-(3) bg
scribedinSectionIIIB,we observethe samebehaviorin
basedonscalinglawsbyspecifyingatrajectoryη(t)=10
the experimentas the maximumnumber ofatoms in the
foralltimestandassumingafixedrelationshipω¯ ∝P1/2
Yb BEC is quite resilient to the evaporation timescale.
[18]. Without these simplifying assumptions,however,it
is essential to turn to numerical techniques, especially
with an additional dynamically controllable parameter
such as CPM amplitude. Therefore, to inform our ex- 5 O
perimental choice of ramp profiles P(t) and h(t), we run opt 40 p
an optimization algorithm in MathematicaTM using nu- γy, 4 tim
merical solutions to equations (2)-(3). To speed up the c iz
n 30 e
numerical integration of N˙ and T˙, we compute the trap e d
apnadramCPetMersaUm(pPli,thu)deanvdalω¯u(ePs.,hW)feorthaendecnosnevgerritdtohfepseowtaer- effici 3 γβopt 20 slop
d 2 opt e
bles into interpolating functions, making the determina- ze , β
etixotnreomfeUly(tf)asatn.d ω¯(t) throughout the evaporation ramp ptimi 1 10 opt (µm
For the functional form of the power ramp P(t) O
0 0 /s
we try single exponential profiles of the form P(t) = )
PP00e−αte/−τPt/τaPn1d+b(i-1e−xpαon)ee−ntt/iaτPl2pr,ogfiuleidseodf tbhyetfhoermfacPt(tt)ha=t 1.0Evapo1r.a5tion tim2.e0scale, 2τ.P5 (s) 3.0
evaporation occurs on an exponential timescale. Fur-
(cid:0) (cid:1)
thermore, a bi-exponential could potentially handle the FIG. 4: CPM amplitude trajectory optimization for various
presence of two dominant timescales (e.g. evaporation powerramptimescalesτP. ForeachvalueτP werunanopti-
and 3-body loss). For the CPM amplitude ramp h(t), mization of the CPM reduction slope β (solid blue triangles)
we try out many different functional forms, including tomaximizetheevaporationefficiencyγ (solidredcircles) at
offset exponential h(t) = h1e−t/τh + (h0 − h1), linear ρ=1.
5
B. Comparison of model and experiment (a) (b)
2 U(t)/10k
B
We experimentally investigate the evaporation effi- ber 107 K) 10 Theory
cwieitnhcyoubrysimmauxliamtiiozninsgwfienfianldBEthCatnouvmerbetrim. eInscaalgerseewmheenret Num 468PSD111000--210 μT (
1-body loss is negligible, the evaporation efficiency is ro- 10-3 1
0 1 2 3 4 5
bust and an exponential reduction of power and linear 0 1 2 3 4 5 0 1 2 3 4 5
(c) (d)
reduction of CPM amplitude yield the largest γ. A typ-
iOcaulrothpetiomreiztiecdalemvaopdoerlastuiocncetsrsafujellcytocrayptiusrsehsotwhendinynFaimg.ic5s. Hz) 100 -1e (s)
omnvineengrt3sf,oorwcreeddewresaviotafp5mo0r0aamgtinosintauftdtoeerainOlloDPwTSDalt.ooaFmdosirnigtnhbeteshefeormweeibnasgeugsrinoe--f ωπ/2 ( 10 Uk/(μK)B 10100 1 2 3 4 5 Loss rat0.1 ΓΓe3vb
the trap to escape. 0 1 2 3 4 5 1 2 3 4 5
ForthemodelcurvesinFig.5,theonlyfreeparameters Evaporation time (s) Evaporation time (s)
are the initial number and temperature, and K . We
3
assume s-wave scattering only as the d-wave threshold
FIG. 5: Example of optimized evaporation trajectory. (a,b)
for 174Yb is 75 µK,and use σ =8πa2, where a=5.6 nm Measured number (blue circles), temperature (red circles),
[26]. Fromaleastsquaresfittothedata,weextractK3 = and phase space density (black circles, inset) evolution dur-
(1.08±0.03)×10−28 cm6s−1. From the behavior of Γev ingforced evaporativecooling showexcellentagreement with
andΓ inFig.5(d), weseethatthe dynamicallyshaped our theoretical model (solid black line, see text). In (b) we
3b
ODT maintains dominance of evaporative over 3-body alsoplotU(t)/10kB (dashedline). Fromthefitin(b)wefind
loss. Furthermore, the elastic collision rate Γel falls less ηavg =10.5. (c)Trajectories oftrapfrequencyanddepth(in-
set)formeasurementsin(a,b). Dashedlinescorrespondtothe
thanafactorof2from1.7kHzto1.1kHzoverthecourse
same trajectories without reduction of CPM amplitude. (d)
of the evaporation sequence. The evaporation efficiency
Evolutionofper-particlelossratesforevaporation(solidline)
γ for this ramp is 3.8, close to the highest value found
and 3-body inelastic loss (dashed line) during forced evapo-
using the optimization algorithm discussed above with
ration.
our initial trap conditions and ramp profiles. We note
that although runawayevaporationwhere dΓ /dt>0 is
el
easilyachievedwiththedynamicallyshapedODT,wedo isviolatedattheseshorttimescales,wecannotapplyour
not find this to be the optimal evaporation strategy due model in this regime.
to enhanced 3-body loss. Fig.6(c) shows absorption images and horizontally in-
tegrated optical density (OD) profiles after the third
evaporation stage for a few final laser powers near the
IV. RAPID BEC PRODUCTION condensation transition. We fit the density profiles to
a bimodal distribution consisting of Gaussian thermal
In addition to providing a platform for highly efficient and Thomas-Fermi BEC profiles. We detect nearly pure
evaporationoflargeatomnumberclouds,dynamicaltrap 174Yb condensates of 1× 105 atoms for Pf = 0.41W,
shapingcanbe appliedto the rapidproductionofBECs. with a total cycle time of 1.8s. By shortening the MOT
For this purpose we shorten the MOT loading and com- loadingandcompressiontime to 0.6s,we produce BECs
pression time to a total of 0.8s, and begin forced evapo- of 0.5×105 atoms with a total cycle time of 1.6s.
rativecooling immediately followingloadingof the ODT
with an initialPSD of <10−4. The experimentally opti-
mizedfastBECrampmeasurementsareshowninFig.6. V. APPLICATION TO LARGE BEC
For fast BEC production, we find the use of three dis- PRODUCTION
tinctevaporationstagestobeoptimal. Inthe firststage,
we exponentially reduce the power by a factor of 20 in We now turn our attention to the production of large
200ms while linearly reducing the CPM amplitude to atom number condensates. As shown in Fig.5(d), the
zeroin150ms. Second,weexponentiallyreducethelaser 3-body loss rate grows noticeably near the point ρ = 1
power by a factor of 6 in 300ms. In a third, relatively duetoalargeincreaseinthedensity. Therefore,inorder
slow evaporation stage, we linearly decrease the power to produce the largest condensates we evaporate with
another 30%in 350ms, and then hold at constant depth the same functional ramps as in Fig.5 until ρ ≈ 1, and
for 150ms. In Fig.6(b) we see that the horizontal and subsequently continue forced evaporation by fixing the
vertical temperatures initially decouple due to the rapid power and increasing the CPM amplitude.
decrease of CPM amplitude in the first 150ms. This Fig.7 shows an example absorption image and inte-
can be understood by considering the adiabatic temper- grated OD profile of a pure BEC produced from such
ature evolution terms, since ω˙ /ω ≫ ω˙ /ω during an evaporation ramp. For this particular measurement
x,z x,z y y
this time. Since the assumption of thermal equilibrium we finish the initial evaporation stage at ρ ≈ 1 with
6
(a) (c)
P= 0.58 W 250
60 f
1Number5×110056 units) 4260000 Pf = 0.52 W ptical Density(arb. units)21150500000
b. 40 O 0
0 ar 20 100 µm -300 -200 -100 0 100 200 300
0 .1 .2 .3 .4 .5 y ( 0 Vertical position (µm)
(b) Evaporation time (s) nsit 60 Pf = 0.46 W
De 40 FIG.7: Production of large, pure174Yb BECs usingdynam-
60
al 20 icaltrapshapingtocombat3-bodyinelasticloss. Absorption
μK) 40 TTVH Optic 600 P= 0.41 W timera5g0e(mlesftT)oaFn,dwhiotrhizfiotnt(aslollyidinltineger)attoedaOTDhopmroafis-leFe(rrmigihtB)EaCf-
T ( 20 40 f density distribution, yielding a total of 1.2×106 condensed
20 100 µm atoms. The thermal fraction is consistent with zero.
0 0
0 .1 .2 .3 .4 .5 -200 0 200
Evaporation time (s) Vertical position (µm) be used to generate multi-well traps.
Ourfastandefficientquantumgasproductionmethods
FIG. 6: Rapid production of 174Yb BECs with a total cy- canbeappliedfruitfullytootheratomicspecies,fermions
cle time of 1.8s. (a,b) Evolution of number and horizontal and mixtures, and can significantly impact various ap-
(TH) and vertical (TV) temperatures during first 2 phases of plications, such as atom interferometry [3, 7, 30]. The
rapid evaporation (see text). (c) Absorption images (insets)
demonstrated fast BEC production time in this work is
and horizontally integrated OD profiles after the 3rd phase
atleastoneorderofmagnitude shorterthanthatin typ-
of evaporation and 25 ms time-of-flight (ToF) to a variable
ical quantum degenerate gas experiments. The leading
final laser power P . Solid lines are fits to a bimodal density
f
limitation to our cycle time is the MOT loading rate,
distribution(seetext),withdashedlinesindicatingfitstothe
stemming from the relatively narrow linewidth of the
thermal component. For P =0.41 W, we find a nearly pure
f
condensate of 1×105 atoms. 1S0 →3P1 transition. Combining broad- and narrow-
line laser cooling either spatially [31] or temporally [32]
canshortenthe cycletime further, andis alsoapplicable
h = 130µm and P = 1.0 W, and subsequently increase to other alkaline-earth [14, 33, 34] as well as lanthanide
the CPM amplitude to h = 180µm. The resulting trap [35–37]atoms. Ourschemeshouldalsoimprovethecycle
frequencies are (ω ,ω ,ω )=2π×(17,110,10)Hz. Fol- timeofalkaliatomexperimentswhichfeaturetheadvan-
x y z
lowingthismethodwecanreliablycreatepure174Ybcon- tage of sub-Doppler cooling with relatively broad tran-
densates of 1.2×106 atoms, a factor of 4 improvement sitions as well as the combination of broad and narrow
over the largest reported Yb BEC number [24], with a transitions [38, 39].
total cycle time of 15s. The broad applicability of our method to all laser-
cooledatomicspeciesgathersadditionalappealwhenone
considersthatmostopticaltrappingexperimentsalready
involve one AOM to control the power during evapora-
VI. SUMMARY AND OUTLOOK
tive cooling. The short cycle time can also allow for fur-
ther technical simplifications including reduced vacuum
WhileanadditionalCPMdegreeoffreedomfromasec-
requirements and the use of lower ODT powers at wave-
ond orthogonal AOM can allow more control over atom
lengths closer to atomic resonance. In alkali atoms such
cloud compression [27] or final BEC shape [28, 29], it is
as Rb and Cs, where laser cooling can produce samples
unlikely that it will provide significant improvements to atafewµKwithPSD>10−3[16,40],anorderofmagni-
the speed and efficiency of evaporative cooling that we
tude lower initial trap depth than used here is adequate.
havedemonstratedhere. Inour currentimplementation, Estimating for the commonly used 87Rb atom, where σ
we choose the vertical waist and large initial CPM am- andK [41]aresimilarto174Yb, itshouldbe possibleto
3
plitudetoloadabout50%ofthecompressedMOT.Once
have 1s BEC production times with only 0.5W of ODT
a large atom number has been loaded, we controlall the
power at 100nm detuning.
relevant parameters for efficient evaporative cooling by
dynamically shaping the trap (Fig.5) to provide an ap-
propriate ω¯ to maximize γ and simultaneously provide
large ω for tight confinement against gravity. Acknowledgments
y
The CPM amplitude provides a simple way to control
the trap aspect ratio over a large range. Furthermore, We thank Chao Yi Chow for various technical contri-
a large CPM amplitude with ω ≫ ω can be used to butions to the experiment. We gratefully acknowledge
y x,z
realize2Dconfinement,andsquare-wavemodulationcan financial support from NSF Grant No. PHY-1306647,
7
(a) (b) Treating ξ˙| ≡ v as a constant determined by ω
ξ=0 0 mod
h) of 012 RRoooott 12 Iof )000..43 h0 = 2.5w0 GDealutas sFiaunnction and h0, we arrive at a differential equation for ξ(t)
ξt() (units --210 Root 3 Ĩx() (units 000...210h0 = 7h.h500 w= =0 5 3w.50 w0 ddξt = 1−(cid:16)v0hξ0(cid:17)2, |ξ|≤h0. (A4)
-1 0 1 -15 -10 -5 0 5 10 15
t (units of ̟/2ω ) x (units of w) Solving equation (A4) gives the implicit equation
mod 0
ξ(t)3/3h2−ξ(t)+v t=0. The3rootsareplottedinFig.
0 0
FIG. 8: FM waveform for parabolic time-averaged potential. 8(a). Clearly the solution that is centered on the y axis
(a)The3rootsoftheimplicitequationξ3/3h2−ξ+t=0plot- willbethedesiredroot. Fromtheconstraint|ξ|≤h ,we
0 0
ted in the domain −π/2ωmod ≤t ≤ π/2ωmod, corresponding find |t| ≤ 2h0/3v0. Anticipating that we will construct
toa half period. The root of interest for CPM is centeredon theperiodicwaveformshownintheinsettoFig. 1(a),we
theξ axis(Root2). (b)Time-averagedCPMpotentialsusing define the modulation period 2π/ω =8h /3v . Next
mod 0 0
root2from(a)astheFMwaveformfordeltafunction(dashed
we compute the time average with t = −2h /3v and
black line) and Gaussian (solid red line) initial beams. 1 0 0
t =2h /3v ,
2 0 0
AFOSR Grant No. FA 9550-15-1-0220,and ARO MURI 2h0/3v0
3v P
Grant No. W911NF-12-1-0476. I˜(x)= 0 0 δ(x−ξ(t))dt
4h
0 −2hZ0/3v0
APPENDIX: FM WAVEFORM DERIVATION π/2ωmod
ω P δ(t−t )
mod 0 0
= dt, (A5)
π |ξ˙(t )|
In order to derive the FM waveform necessary to cre- −π/2Zωmod 0
ate the parabolic time-averaged potential shown in Fig.
1(b), weworkwith 1Dtransverseprofilesinthe painting
where t satisfies x−ξ(t ) = 0. From equation (A4) we
dimension, x. Consider the unmodulated beam shape to 0 0
have ξ˙(t ) = v /(1−(ξ(t )/h )2). Lastly, the integral
be a delta function I(x) = P δ(x). In this case, for a 0 0 0 0
0
over the delta function evaluates to zero unless |t | ≤
CPM amplitude of h we want the time-averaged inten- 0
0
2h /3v , which is equivalent to |ξ(t )|≤h . Therefore,
sity distribution to be of the form 0 0 0 0
I˜(x)= 34Ph300(h20−x2)Θ(h0−|x|), (A1) I˜(x)= 34Ph00 1−(ξ(t0)/h0)2 Θ(h0−|ξ(t0)|)
3P (cid:0) (cid:1)
where Θis the Heavisidefunction. To realizethis poten- = 0 h2−x2 Θ(h −|x|). (A6)
4h3 0 0
tial we want to find the function ξ(t) such that the time 0
(cid:0) (cid:1)
averageof I(x−ξ(t)) from t to t equals the expression
1 2
in equation (A1), where ξ(t1)=−h0 and ξ(t2)=h0. Fig. 8(b) shows the time-averaged potentials for
Wereasonthattherastereddeltafunctionmustspend delta function I(x) = P δ(x) and Gaussian I(x) =
′ 0
an amount of time at each point ξ that obeys I exp(−2x2/w2) initial beam shapes, where I =
0 0 0
2/πP /w . Asseeninthe figure,the casesofthe delta
dt|ξ=ξ′ I˜(ξ′) ξ′ 2 function0 an0d Gaussian differ very little, and become in-
dt|ξ=0 = I˜(0) = 1−(cid:18)h0(cid:19) !. (A2) dpiscernible when h0 is much greater than the Gaussian
waist. Inordertoperformfrequencymodulationwiththe
Writing dt as dξ/ξ˙ we find waveform ξ(t) corresponding to the second root in Fig.
8(a), we utilize the arbitrary waveform functionality of
dt|ξ=ξ′ ξ˙|ξ=0 ξ′ 2 a Stanford Research Systems DS345 function generator,
= = 1− . (A3) passing it an array of values very closely approximating
dt|ξ=0 ξ˙|ξ=ξ′ (cid:18)h0(cid:19) ! the periodic continuation of ξ(t) (see inset to Fig. 1).
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